logo AFST

Méthodes de changement d’échelles en analyse complexe
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 3, pp. 427-483.

Nous mettons en perspective différentes méthodes de changement d’échelles et illustrons leur pertinence en mettant sur pieds des preuves simples et élémentaires de plusieurs théorèmes biens connus en analyse ou géométrie complexe. Les situations abordées sont variées et la plupart des théorèmes démontrés sont des classiques initialement obtenus entre la fin du xixe  et la seconde moitié du xxe  siècle.

We discuss several rescaling methods in complex analysis and geometry and apply them to get elementary proofs of some classical results. The Bloch principle plays an important role in our approach and yields to a somewhat unified point of view.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1127
@article{AFST_2006_6_15_3_427_0,
     author = {Fran\c{c}ois Berteloot},
     title = {M\'ethodes de changement d{\textquoteright}\'echelles en analyse complexe},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {427--483},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {6e s{\'e}rie, 15},
     number = {3},
     year = {2006},
     doi = {10.5802/afst.1127},
     zbl = {1123.37019},
     mrnumber = {2246412},
     language = {fr},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1127/}
}
François Berteloot. Méthodes de changement d’échelles en analyse complexe. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 3, pp. 427-483. doi : 10.5802/afst.1127. https://afst.centre-mersenne.org/articles/10.5802/afst.1127/

[1] V. Arnold Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Editions MIR Moscou - Editions du Globe Paris, 1996 | MR 898218 | Zbl 0455.34001

[2] D. Bargmann Simple proofs of some fundamental properties of the Julia set, Ergodic Theory Dynam. Systems, Volume 19 (1999) no. 3, pp. 553-558 | MR 1695942 | Zbl 0942.37033

[3] E. Bedford; S. Pinchuk Domains in C 2 with non-compact automorphism group, Math. USSR Sbornik, Volume 63 (1989), pp. 141-151 | MR 937803 | Zbl 0668.32029

[4] E. Bedford; S. Pinchuk Convex domains with non-compact automorphism group, J. Geometric Anal., Volume 1 (1991), pp. 165-191 | MR 1120679 | Zbl 0733.32014

[5] S. Bell The Bergman kernel function and proper holomorphic mappings, Trans. Amer. Math. Soc., Volume 270 (1982) no. 2, pp. 685-691 | MR 645338 | Zbl 0482.32007

[6] S. Bell; E. Ligocka A simplification and extension of Fefferman’s theorem on biholomorphic mappings, Invent. Math., Volume 57 (1980) no. 3, pp. 283-289 | MR 568937 | Zbl 0411.32010

[7] W. Bergweiler Rescaling principles in function theory, Analysis and its applications (2001), pp. 11-29 ((Chennai, 2000)) | MR 1893221 | Zbl 0993.30018

[8] F. Berteloot Attraction de disques analytiques et continuité Höldérienne d’applications holomorphes propres, Topics in Compl.Anal., Banach Center Publ. (1995), pp. 91-98 | MR 1341379 | Zbl 0831.32012

[9] F. Berteloot Characterization of models in C 2 by their automorphism group, Int. J. Math., Volume 5 (1994) no. 5, pp. 619-634 | MR 1297410 | Zbl 0817.32010

[10] F. Berteloot Principe de Bloch et estimations de la métrique de Kobayashi des domaines de C 2 , J. Geom. Anal., Volume 13 (2003) no. 1, pp. 29-37 | MR 1967034 | Zbl 1040.32011

[11] F. Berteloot; G. Cœuré Domaines de C 2 , pseudoconvexes et de type fini ayant un groupe non-compact d’automorphismes, Ann. Inst. Fourier Grenoble, Volume 41 (1991) no. 1, pp. 77-86 | EuDML 74919 | Numdam | Zbl 0711.32016

[12] F. Berteloot; C. Dupont Une caractérisation des exemples de Lattès par leur mesure de Green (Comment. Math. Helv. (à paraître)) | Zbl 1079.37039

[13] F. Berteloot; J. Duval Une démonstration directe de la densité des cycles répulsifs dans l’ensemble de Julia, Complex analysis and geometry, Volume 188, Progr. Math., Basel, 2000, p. 221-222 ((Paris, 1997)) | Zbl 1073.37522

[14] F. Berteloot; J. Duval Sur l’hyperbolicité de certains complémentaires, L’Enseignement Mathématique, Volume 47 (2001), pp. 253-267 | MR 1876928 | Zbl 1009.32015

[15] F. Berteloot; J. J. Loeb Une caractérisation géométrique des exemples de Lattès de P k , Bull. Soc. Math. Fr., Volume 129 (2001) no. 2, pp. 175-188 | EuDML 272412 | Numdam | MR 1871293 | Zbl 0994.32026

[16] F. Berteloot; V. Mayer Rudiments de dynamique holomorphe (Cours Spécialisés) Volume 7, Société Mathématique de France, EDP Sciences, Les Ulis, 2001 | MR 1973050 | Zbl 1051.37019

[17] J.-Y. Briend; J. Duval Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de P k , Acta Math., Volume 182 (1999) no. 2, pp. 143-157 | MR 1710180 | Zbl 01541209

[18] J.-Y. Briend; J. Duval Deux caractérisations de la mesure d’équilibre d’un endomorphisme de P k , Publ. Math. Inst. Hautes Études Sci., Volume 93 (2001), pp. 145-159 | EuDML 104174 | Numdam | MR 1863737 | Zbl 1010.37004

[19] R. Brody Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc., Volume 235 (1978), pp. 213-219 | MR 470252 | Zbl 0416.32013

[20] J. Byun; H. Gaussier; K.-T. Kim Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group, J. Geom. Anal., Volume 12 (2002) no. 4, pp. 581-599 | MR 1916860 | Zbl 1039.32003

[21] D. Catlin Estimates of Invariant metrics on pseudoconvex domains of dimension two, Math. Z., Volume 200 (1989), pp. 429-466 | EuDML 174019 | MR 978601 | Zbl 0661.32030

[22] M. Christ C irregularity of the ¯-Neumann problem for worm domains, J. Amer. Math. Soc., Volume 9 (1996) no. 4, pp. 1171-1185 | MR 1370592 | Zbl 0945.32022

[23] B. Coupet Precise regularity up to the boundary of proper holomorphic mappings, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume (4) 20 (1993) no. 3, pp. 461-482 | EuDML 84157 | Numdam | MR 1256077 | Zbl 0812.32011

[24] B. Coupet; H. Gaussier; A. Sukhov Regularity of CR maps between convex hypersurfaces of finite type, Proc. Amer. Math. Soc., Volume 127 (1999) no. 11, pp. 3191-3200 | MR 1610940 | Zbl 0951.32026

[25] B. Coupet; S. Pinchuk; A. Sukhov On boundary rigidity and regularity of holomorphic mappings, Int. J. Math., Volume 7 (1996) no. 5, pp. 617-643 | MR 1411304 | Zbl 0952.32011

[26] B. Coupet; A. Sukhov On CR mappings between pseudoconvex hypersurfaces of finite type in C 2 , Duke Math. J. Vol., Volume 88 (1997) no. 2, pp. 281-304 | MR 1455521 | Zbl 0895.32007

[27] J.-P. Demailly Variétés hyperboliques et équations différentielles algébriques, Gaz. Math., Volume 73 (1997), pp. 3-23 | MR 1462789 | Zbl 0901.32019

[28] J.-P. Demailly; J. ElGoul Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math., Volume 122 (2000) no. 3, pp. 515-546 | MR 1759887 | Zbl 0966.32014

[29] G. Dethloff; G. Schumacher; P. M. Wong On the hyperbolicity of the complements of curves in algebraic surfaces, Duke Math. J., Volume 78 (1995), pp. 193-212 | MR 1328756 | Zbl 0847.32028

[30] K. Diederich; J. E. Fornaess Proper holomorphic maps onto pseudoconvex domains with real analytic boundary, Ann. Math., Volume 110 (1979), pp. 575-592 | MR 554386 | Zbl 0394.32012

[31] K. Diederich; S. Pinchuk Proper holomorphic maps in dimension 2 extend, Indiana. Math. J., Volume 44 (1995), pp. 1089-1126 | MR 1386762 | Zbl 0857.32015

[32] T. C. Dinh; N. Sibony Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9), Volume 82 (2003) no. 4, pp. 367-423 | MR 1992375 | Zbl 1033.37023

[33] P. G. Dixon; J. Esterle Michael’s problem and the Poincaré-Fatou-Bieberbach phenomenon, Bull. Amer. Math. Soc. (N.S.), Volume 15 (1986) no. 2, pp. 127-187 | MR 854551 | Zbl 0608.32008

[34] A. M. Efimov A generalization of the Wong-Rosay theorem for the unbounded case, Sb. Math., Volume 186 (1995) no. 7, pp. 967-976 | MR 1355455 | Zbl 0865.32020

[35] A. Eremenko A Picard type theorem for holomorphic curves, Period. Math. Hungar., Volume 38 (1999) no. 1-2, pp. 39-42 | MR 1721476 | Zbl 0940.32010

[36] C. Fefferman The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., Volume 26 (1974), pp. 1-65 | EuDML 142293 | MR 350069 | Zbl 0289.32012

[37] J. E. Fornaess; N. Sibony Construction of P.S.H. functions on weakly pseudoconvex domains, Duke Math. J., Volume 58 (1989), pp. 633-656 | MR 1016439 | Zbl 0679.32017

[38] J. E. Fornaess; N. Sibony Complex Dynamics in higher dimensions, Complex potential theory (Montréal, PQ, 1993) (NATO ASI series Math. and Phys. Sci.) Volume 439 (1994), pp. 131-186 | MR 1332961 | Zbl 0811.32019

[39] J. E. Fornaess; N. Sibony Complex Dynamics in higher dimensions II, Ann. of Math. Studies, Volume 137 (1995) (Princeton Univ. Press, Princeton, NJ) | MR 1369137 | Zbl 0847.58059

[40] F. Forstneric An elementary proof of Fefferman’s theorem, Exposition. Math., Volume 10 (1992) no. 2, pp. 135-149 | MR 1164529 | Zbl 0759.32018

[41] F. Forstneric; J.-P. Rosay Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann., Volume 110 (1987), pp. 239-252 | EuDML 164320 | MR 919504 | Zbl 0644.32013

[42] J.-P. Françoise Géométrie analytique et systèmes dynamiques, Presses Universitaires de France, Paris, 1995 (Cours de troisième cycle) | MR 1620294

[43] S. Frankel Complex geometry of convex domains that cover varieties, Acta Math., Volume 163 (1989) no. 1-2, pp. 109-149 | MR 1007621 | Zbl 0697.32016

[44] H. Gaussier Characterization of models for convex domains (preprint) | MR 1460422

[45] I. Graham Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in C n with smooth boundary, Trans. Amer. Math. Soc., Volume 207 (1975), pp. 219-240 | MR 372252 | Zbl 0305.32011

[46] M. Gromov Foliated plateau problem, part II : harmonic maps of foliations, GAFA, Volume 1 (1991) no. 3, pp. 253-320 | EuDML 58124 | MR 1118731 | Zbl 0768.53012

[47] G. Henkin An analytic polyhedron is not biholomorphic to a strictly pseudoconvex domain, Dokl. Akad. Nauk SSSR, Volume 210 (1973), pp. 1026-1029 | MR 328125 | Zbl 0288.32015

[48] A. Isaev; S. Krantz Domains with non-compact automorphism group : a survey, Adv. Math., Volume 146 (1999) no. 1, pp. 1-38 | MR 1706680 | Zbl 1040.32019

[49] M. Jonsson; D. Varolin Stable manifolds of holomorphic diffeomorphisms, Invent. Math., Volume 149 (2002) no. 2, pp. 409-430 | MR 1918677 | Zbl 1048.37047

[50] K.-T. Kim; S. Krantz Some new results on domains in complex space with non-compact automorphism group, J. Math. Anal. Appl., Volume 281 (2003) no. 2, pp. 417-424 | MR 1982663 | Zbl 1035.32019

[51] G. Kobayashi Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, Volume 318, Springer-Verlag, Berlin, 1998 | MR 1635983 | Zbl 0917.32019

[52] E. Landau Uber di Blochste Konstante und zwei verwandte Weltkonstanten, Math. Z., Volume 30 (1929), pp. 608-634 | EuDML 168142 | JFM 55.0770.03 | MR 1545082

[53] S. Lang Introduction to complex hyperbolic manifolds, Springer Verlag, 1987 | MR 886677 | Zbl 0628.32001

[54] E. Ligocka Some remarks on extension of biholomorphic mappings, Analytic functions (Lecture Notes in Math.) (1980), pp. 350-363 | MR 577466 | Zbl 0458.32008

[55] A. J. Lohwater; Ch. Pommerenke On normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I (1973) no. 550 | MR 338381 | Zbl 0275.30027

[56] S. Pinchuk On proper holomorphic mappings of strictly pseudoconvex domains, Sib. Math. J., Volume 15 (1974), pp. 909-917 | Zbl 0303.32016

[57] S. Pinchuk The scaling method ans holomorphic mappings, Proc. Sympos. Pure Math., Volume 52 (1991) no. 1, pp. 151-161 | Zbl 0744.32013

[58] S. Pinchuk; S. Khasanov Asymptotically holomorphic functions and their applications, Math. USSR-Sb., Volume 62 (1989) no. 2, pp. 541-550 | MR 933702 | Zbl 0663.32006

[59] R. Range Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, Volume 108, Springer-Verlag, New York, 1986 | MR 847923 | Zbl 0591.32002

[60] A. Ros The Gauss map of minimal surfaces, Differential Geometry-Valencia (2001), pp. 235-250 | MR 1922054 | Zbl 1028.53008

[61] J.-P. Rosay Sur une caractérisation de la boule parmi les domaines de C n par son groupe d’automorphismes, Ann. Inst. Fourier, Volume 29 (1979) no. 4, pp. 91-97 | EuDML 74435 | Numdam | MR 558590 | Zbl 0402.32001

[62] J.-P. Rosay; W. Rudin Holomorphic maps from C n to C n , Trans. Amer. Math. Soc., Volume 310 (1988) no. 1, pp. 47-86 | MR 929658 | Zbl 0708.58003

[63] D. Ruelle Elements of differentiable dynamics and bifurcation theory, Academic Press, Inc., Boston, MA, 1989 | MR 982930 | Zbl 0684.58001

[64] W. Schwick Repelling periodic points in the Julia set, Bull. London Math. Soc., Volume 29 (1997) no. 3, pp. 314-316 | MR 1435565 | Zbl 0878.30020

[65] N. Sibony A class of hyperbolic manifolds, Ann. Math. Studies (1981) no. 100, pp. 357-372 | MR 627768 | Zbl 0476.32033

[66] N. Sibony Dynamique des applications rationnelles de P k , Panor. Synthèses (1999), pp. 97-185 | MR 1760844 | Zbl 1020.37026

[67] Y. T. Siu; S. K. Yeung Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane, Invent. Math., Volume 124 (1996) no. 1-3, pp. 573-618 | MR 1369429 | Zbl 0856.32017

[68] B. Stensones A proof of the Michael conjecture (1998) (Preprint)

[69] B. Stensones Fatou-Bieberbach domains with C -smooth boundary, Ann. of Math. (2), Volume 145 (1997) no. 2, pp. 365-377 | MR 1441879 | Zbl 0883.32020

[70] S. Sternberg Local contractions and a theorem of Poincaré, Amer. J. Math., Volume 79 (1957), pp. 809-824 | MR 96853 | Zbl 0080.29902

[71] A. Sukhov On boundary regularity of holomorphic mappings, Mat. Sb., Volume 185 (1994), pp. 131-142 | MR 1317303 | Zbl 0843.32016

[72] S. Webster On the reflection principle in several complex variables, Proc. Amer. Math. Soc., Volume 71 (1978) no. 1, pp. 26-28 | MR 477138 | Zbl 0626.32019

[73] B. Wong Characterization of the unit ball in C n by its automorphism group, Invent. Math., Volume 41 (1977) no. 3, pp. 253-257 | EuDML 142493 | MR 492401 | Zbl 0385.32016

[74] L. Zalcman Normal families : new perspectives, Bull. Amer. Math. Soc., Volume 35 (1998), pp. 215-230 | MR 1624862 | Zbl 1037.30021

[75] L. Zalcman A heuristic principle in complex function theory, Amer. Math. Monthly, Volume 82 (1975) no. 8, pp. 813-817 | MR 379852 | Zbl 0315.30036